Download Applications of Algebraic Geometry to Coding Theory, Physics by I. Burban, Yu. Drozd, G.-M Greuel (auth.), Ciro Ciliberto, PDF

By I. Burban, Yu. Drozd, G.-M Greuel (auth.), Ciro Ciliberto, Friedrich Hirzebruch, Rick Miranda, Mina Teicher (eds.)

An updated file at the present prestige of significant examine issues in algebraic geometry and its functions, reminiscent of computational algebra and geometry, singularity conception algorithms, numerical recommendations of polynomial structures, coding idea, conversation networks, and machine imaginative and prescient. Contributions on extra primary facets of algebraic geometry contain expositions regarding counting issues on kinds over finite fields, Mori thought, linear structures, Abelian types, vector bundles on singular curves, degenerations of surfaces, and replicate symmetry of Calabi-Yau manifolds.

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Example text

And if the general curve in the system is irreducible. then it has ordinary singular points ofmultiplicifies ml, ... ,mll at PI, ""PI1 and it has no other singularities. £ corresponding to systems £d(p~ll " .. ,p~lll) of plane curves. a (pi;! • , p;:lll) is special if and only if it is ( -1) -special. 4. Segre Implies Harbourne-Hirschowitz Note that Harbourne-Hirschowitz Conjecture immediately implies Segre's Conjecture, since (- 1)-special systems are not reduced. In this section we will show that in fact Segre's Conjecture is not weaker that the Harbourne-Hirschowitz.

In the following S = K[xo, ... ,XII] is a ring of polynomials over a field K, K is the algebraic closure of K and the characteristic of K is not 2. By projective algebraic variety we mean the set of points ofJIDj; that are the zeros of the polynomials of a homogeneous ideal of S. In particular, by a (projective) curve in JIDA- we mean a projective variety of pure dimension 1. 2. Maximal rank and minimal generation of disjoint unions of non special curves We will use freely the common notations of sheaf cohomology in JIDII.

Ciliberto et al. ), Applications of Algebraic Geometry to Coding Theory, Physics and Computation, 37-51. © 2001 Kluwer Academic Publishers. 38 C. CILIBERTO AND R. MIRANDA the on()' way speciality can arise (see Some authors, among them Gimigliano (1987), also pointed out an earlier conjecture, formulated by B. Segre (1961), to the effect that if a system is special then it has some multiple curve in its base locus (see §2). Clearly Harbourne-Hirschowitz conjecture implies Segre's one. In this paper we prove that also the converse holds (see §4).

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